A PROOF OF THE TWIN PRIME THEOREM

Paul J. Brinson Kenmore, New York

2011

OUTLINE OF THE PROOF

1) Let P = any prime, n ≡ the ordinal number of a prime, and Pn = the nth prime.

2) Let P!! or Pn!! ≡ primorial P or the product of the first n primes.

3) A “factor” of an integer, is another integer, other than the first integer, which

leaves a remainder of zero when dividing the first integer.

4) Let p = any prim, a prim being any integer from zero though Pn!! which does not

have any of the n prime factors of Pn!! as a factor of the prim itself.

5) Let (tp) or (p1, p2) be any ordered pair of twin prims from zero through Pn!! +1, of the form (p, p +2), the last pair, (Pn!! – 1, Pn!! + 1), being called the straddle pair. 6) Let [P!!] ≡ the ordered set of integers from zero through Pn!!, or the number of

them, and [P] be the ordered set of primes from zero through Pn!!, or the number

of them.

7) Let [p] or [p]n ≡ the ordered set of prims from zero through Pn!!, or the number of

them.

8) Let [tp] or [tp]n ≡ the ordered set of twin prim pairs from 0 through Pn!!, or the

number of them, but also including the straddle pair, (Pn!! - 1, Pn!! + 1).

9) Let [tp]x ≡ the ordered set of twin prim pairs in [Pn!!] from 0 through x, or the

number of them, where x is an integer from zero through Pn

10) Prove that [tp]n = (P1 - 1)(P2 - 2) (P3 -2)((P4 - 2)(P5 - 2)(P6 - 2)(P7 – 2) - -….(Pn –2),

the number of twin prim pairs in [Pn!!], provided that the straddle pair, (Pn!! - 1, Pn!!

+ 1) is defined as being “in” [Pn!!], even though the second member of the straddle pair is

greater that Pn!!. A symbol for the formula for [tp]n is “(Pn – 2)!!”, and n may increase

without upper bound,

11) The ratio of the number of prims in [Pn!!] to Pn!! is called the prim ratio for Pn!!

and equals [p]n / Pn!!. The twin prim ratio for Pn!! is [tp]n / Pn!!.

12) P2 ≡ P^2 when P2 is written as a subscript.

13) [p]x/x is called the prim ratio through x in [Pn!!], where x is an integer.

14) The prim ratio through (Pn)2 in [Pn!!] = [p]P^2 / (Pn

2).

15) Demonstrate that all of the prims in [Pn!!] less than (Pn)2, except “1”, are primes.

16) Prove that where Pn > 1069, the number of twin prim and twin prime pairs

between Pn and (Pn)2 are always more than“8Pn,” for all values of Pn, without upper limit.

17) Ergo, there is an infinite number of twin prime pairs.

POINT 1

DISCUSSION OF PRIMS AND PRIMORIAL NUMBERS 1) Let P = any prime, n = the ordinal number of any prime and Pn = the nth prime.

(P1 = 2, P2 = 3, P3 =5, P4 = 7, P5 = 11, P6 = 13, ….P10 = 29 …. ad infinitum.

2) Let Pn!! = P1 x P2 x P3 x P4 xP5 x P6 x P7 x P8 ….Pn –1 x Pn, the product of the first n

primes, and let P!! be identical to Pn!!, each being read “ P-subn primorial” or “pri-

morial P-subn. (P1!! = 2, P2!! = 2 x 3 = 6, P3!! = 2 x 3 x 5 = 30, P4!! = 2 x 3 x 4 x 7 = 210,

P5!! = 2 x 3 x 5 x 7 x 11 = 2310, P6!! = 2 x 3 x 5 x 7 x 11 x 13 = 30,030, ….)

3) [Pn!!] represents the ordered set of integers from zero through Pn!!, and [P!!] is

identical to [Pn!!] when it is understood that P is Pn.

4) A prim is any integer less than Pn!!, which does not have any of the prime factors

of Pn!! as a factor of itself. “p” represents any prim and [p]n, or [p] with no subscript,

represents the ordered set of prims from zero to Pn!!, or the number of them, and is

known as the prim set for Pn!!, or the prim set for P-subn primorial. According to

definition, “1” is a prim in each prim set, but the prime factors of Pn!! are not.

5) A nonprim is any integer from zero through Pn!! which does have at least one of

the prime factors of Pn!! as a factor of itself. “m” represents any nonprim and [m]n or [m],

without a subscript, represents the ordered set of nonprims from zero through Pn!!, or the

number of them, and is known as the nonprim set for P-subn primorial. By definition,

each of the prime factors of Pn!! is a nonprim. The sum of the number of prims and

number of nonprims from zero through Pn!! is Pn!!. ([m] + [p] = Pn!! )

7) The first five prim sets are as follows:

i) [p] for 2!! = (1), where 2!! = 2

ii) [p] for 3!! = (1, 5), where 3!! = 3x2 = 6.

iii) [p] for 5!! = (1, 7, 11, 13, 17, 19, 23, 29), where 5!! = 5x3x2 =30.

iv) [p] for 7!! = (1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73,

79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157,

163, 167, 169, 173, 179, 181, 187, 189, 191, 197, 199, 209) where 7!! = 7x5x3x2

=210.

v) [p] for 11!! = (1, 13, 17, 19, 23 …. 2310 – 13, 2310-1) where 11!! = 2,310.

8) Each prim set, except the first, for a given primorial number, Pn!!, may be

constructed by taking the prims of the last prior prim set, the prim set for Pn-1!!, and

adding each of said prims, separately and in numerical order to zero, and also to each

of the integral multiples of Pn-1!! from “1” to Pn –1 separately and in numerical order,

to form what is called the “preprim” set for Pn!!, and then,

9) to complete the construction of the prim set for Pn!! from the preprim set for Pn!!, a

set of prims called the “strike set” is stricken from the preprim set, leaving the

remaining preprims as the prim set for Pn!! (see illustration infra.).

10) The strike set, [ss], for the preprim set for Pn!!, consists of all of the members of

the preprim set in numerical order which have Pn as a factor, or the number of them.

The result is that none of the members of the set of prims left after removing the

strike set has any of the n prime factors of Pn!! as a factor if itself, and therefore the

result is the prim set for Pn!!.

11) The number of members of the strike set for the preprim set of Pn!! equals the

number of members of the prim set for Pn-1!!.

12) The members of the strike set are constructed by multiplying Pn times each of the

prims of the prim set for Pn-1!!, separately, and [ss] = members of the ordered strike

set, or the number of them.

13) None of the members of the preprim set for Pn!! have any of the factors of Pn-1!! as factors of themselves because each of the members of the preprim set are of the form N(Pn-1!!) + m, where the first term has all of the prime factors of Pn-1!! and the second term has none of the prime factors of Pn-1!!. The preprim set for Pn contains all of the

integers less than Pn!! that do not have any of the prime factors of Pn-1 because all of The other integers from zero through Pn!!, the nonpreprims, are of the form N(Pn-1!!) + m, where the first term and second term do have at least one of the prime factors of Pn-1!! in common. This means that to eliminate all of the members of the preprim set for Pn!! which have Pn as a factor, one need only remove the strike set from said preprim set. All of the members of the strike set must be members of the preprim set for Pn!! because they cannot be members of the nonpreprim set for Pn!!, each of which has at least one of the factors of Pn-1!!. 14) The first member of the prim set is always “1” and the last is always Pn!! -1.

Tables for prim sets for P2!! or 3!!, which is 3x2 or 6; P3!! Or 5!!, which is 5x3x2 or 30!! and P4!! which is 7x5x3x2 or 210 are expressed below with each leftmost column consisting of the ordered set, from top down, of Pn terms from zero times

Pn – 1!! through all of the integral multiples of said primorial number from “1” to

Pn – 1. The top row consists of the ordered set of prims for Pn –1!! The set of numbers

at the intersections of the columns and rows constitute the preprim set for Pn!!; the

set of integers which are underlined is the strike set, [ss], from the preprim set; and

the integers not underlined constitute the prim set for Pn!!.

PREPRIM AND PRIM SETS FOR P2!!, OR 3!!

10x2 11x2 32x2 5

(1, 3, 5) is the preprim set for 3!!.

(3) is the strike set for said preprim set.

(1, 5) is the prim set for 3!!.

PREPRIM AND PRIM SETS FOR P3!!, OR 5!!

1 5 0X6 1 5 1X6 7 11 2X6 13 17 3X6 19 23 4X6 25 29

(1, 5, 7, 11, 13, 17,19, 13, 25, 29) is the preprim set for 5!!.

(5, 25) is the strike set for said prim set.

(1, 7, 11, 13, 17, 19, 23, 29) is the prim set for 5!! or, 30.

There are 8 prims in the set and the prim set for 5!! is the largest prim set in which all

of the prims, except “1” are primes. There are 22 nonprims from zero through thirty.

PREPRIM AND PRIM SETS FOR P4!!, OR 7!!

1 7 11 13 17 19 23 29 0x30 1 7 11 13 17 19 23 29 30 31 37 41 43 47 49 53 59 60 61 67 71 73 77 79 83 89 90 91 97 101 103 107 109 113 119 120 121 127 131 133 137 139 143 149

150 151 157 161 163 167 169 173 179 180 181 187 191 193 197 199 203 209 The integers in the above table constitute the preprim set for 7!!.

The eight integers (7x1 = 7, 7x7 = 49, 7x11 = 77, 7x13=91. 7x17 = 119, 7x19 =133, 7x

23 = 161, 7x29 = 203) constitute the strike set for 7!!.

The prim set for 7!! consists of the 48 integers in the above chart which are not

underlined, or the preprim set minus the strike set.

The set: (1, 11x11= 121, 11x13 = 143, 13x 13 =169) are the prims in the set which

consists of prims which are not primes.

Similar tables can be made for all of the subsequent prim and nonprim sets, except

that they become so large that they can be handled only by computer or the imagination

Excepting the first prim set, all prim sets are symmetrical about Pn!! / 2, because for

each prim “p” in [Pn!!], “Pn-p” is also a prim. Such pair of prims are said to be a

“symmetrical” pair of prims and to be complementary one another, as their sum is Pn!!.

The prims in a prim set, as a whole, are bilaterally symmetrical, as for each value of p,

p/2 + (P!! -p)/2 = P!!/2.

.

The preprim sets and the strike sets are also bilaterally symmetrical. The nonprim

sets are also bilaterally symmetrical except that, because zero is neither a prim nor a

nonprim, Pn!! has no other nonprim to make a symmetrical pair.

In those systems in which “1” is considered a prime, “1” may be represented by, “P0.”

“0” = [P0!!].

POINT 2

THE FORMULA FOR TH NUMBER OF PRIMS IN A PRIM SET, [p]n, IS: [p]n ≡ (Pn – 1)!! = (P1 –1)(P2 –1)(P3 –1)(P4 –1)(P5 –1) …..(Pn-2 –1)(Pn-1 –1)(Pn –1).

By inspection we note that this formula works for the first four primorial numbers:

1) [p]1 = 2 - 1 = 1, “1” being the only prim for 2!!.

2) [p]2 = (2 - 1)(3 - 1) = (1)(2) = 2, the number of prims for 3!!, the set of prims being (1, 5).

3) [p]3 = (2 - 1)(3 - 1)(5 - 1) = (1)(2)(4) = 8, the number of prims for 5!!,.

the set of prims being (1, 7, 11, 13, 17, 19, 23, 29).

4) [p] 4 = (2 - 1)(3 - 1)(5 - 1)(7 –1) = (1)(2)(4)(6) = 48, the same as the number

of prims shown in the prim set for 7!! illustrated in Point 1.The title formula

is called the “prim formula”, and, as was shown as in Point 1, the number of

prims in a prim set increases, without upper bound as each successive value

of “n” increases, without upper bound.

5) To obtain the number of prims in [p]5 or any larger [p]n:

a) Multiply the number of prims in [p]4 by P5 to obtain (P5)([p]4). the number of

preprims in the preprim set for P5, and to obtain the number of preprims in the

preprim set for [p]n, multiply the number of prims in [p]n – 1 by Pn to obtain

(Pn)([p]n – 1).

b) To obtain the number of the number of prims in [p]5 and [p]n, subtract the number of

strikes in the strike set for [p]5 and [p]n, [p]4 and [pn – 1] from the number of prims

in the preprim sets for [p]5 and [p]n respectively, to obtain, (P1-1)(P2-1)(P3 – 1)(P4

– 1)(P5 – 1) …. (Pn – 1 – 1)(Pn – 1), which is the title formula and good for all

positive values of n without upper bound.

POINT 3

DISCUSSION OF TWIN-PRIMS AND TWIN- PRIM SETS

1) A twin-prim pair is any two prims of the form (p, p + 2), at least one of whose

members is in the set of prims for [Pn!!];

2) A prim pair of the form (Pn!! - 1, Pn!! +1) is called the straddle pair for [Pn:], and

only the first member of the pair is a member of that set. The first few straddle

pairs for the first several prim sets are: (1, 3), (5, 7), (29,31), (209, 211), ( 2,309,

2,311) … Straddle prim pairs are considered to be “in” [Pn!!];

3) A prim pair of the form (1, Pn!!) is called a polar or cyclical pair for [Pn!!]. Some

examples are: (1, 1), (1, 7), (1, 29), (1, 209), and (1, 309);

4) Alternate ways of dealing with the straddle pair are: (a) to consider the straddle

pair to be “in” [Pn!! + 1]; (b) to consider the straddling pair for Pn!! to be in [Pn+1!!];

or (c) to consider the first member of the straddle pair of prims to be ½ of a twin prim

pair and “1” which is in each prim set to be 1/2 of the twin-prim pair; however, for

this proof, the treatment of straddle pairs will be considered to be “in” Pn!!.

4) There is one twin-prim pair “in [2!!], (1, 3) (Note that “3”, otherwise, is never

appears in a prim set.

5) There is one twin-prim pair in [3!!], (5, 7).

6) There are three twin-prim pairs in [5!!]. (11, 13), (17, 19), (29, 31).

7) There are 15 twin-prim pairs in [7!!]: (11, 13). (17, 19). (29, 31), (41, 43), (59,

61), (101, 103), (107, 109), (137, 139), (149, 151), (167, 169) (179, 181), (191,

193), (197, 199), (209, 211).

8) There are 135 twin-prim pairs in [11!!], or [2310], which can be found by

inspection.

9) The numbers of twin-prim pairs in each of the above prim sets correspond to the

numbers given by the formula for the number of twin-prim pairs in a prim set in

the “Outline of the Proof”: [tp] = (P1 -1)(P2 - 2)(P3 – 2)(P4 – 2)(P5 - 2) ….(Pn – 2)

(Pn – 2)!!.

10) Each twin-prim set, except the first, for a given primorial number, Pn!!, may be

constructed by taking the set of twin-prim pairs for the last prior prim set, the twin

prim set for [Pn-1!!], and adding the members of each of said prim-pairs of said

set, separately and in numerical order to zero, and also to each of the integral

multiples ofPn-1!! from 1 to Pn –1, separately and in numerical order to form what

is called the pre-twin-prim set for [Pn!!],which contains Pn times the number of

twin-prim pairs in [Pn-1!!]. To complete the construction of the twin-prim set for

Pn!!, a set of twin-prim pairs called the “twin-prim strike set” is stricken from the

pre-prim set for [Pn!!]. The number of twin-prim pairs stricken by striking the

twin-prims from the pre-twin-prim set is 2 times the number of pre-twin-prim

pairs in the twin-prim set for [Pn-11!!], so that the number of twin-prim pairs left

in the twin-prim set for [Pn!!] equals (Pn –2) times the number of twin-prim pairs

in the twin-prim pair set for [Pn-1!!],

POINT 4 IF “n” IS ANY INTEGER, WITHOUT UPPER LIMIT, THE FORMULA FOR

THE NUMBER OF TWIN PRIM PAIRSIN THE PRIM SET FOR [Pn!!] IS: [tp]n = (P1 – 1)(P2 – 2)(P3 –2)(P4 – 2)(P5 –2) ……(Pn –2) (Pn – 2)!!.

Let:

1) “tp” = any twin prime pair;

2) [pt]n = the ordered set of twin prim pairs in (Pn – 1)!!, or the number of them;

3) (Pn –2)!! (P1 – 1)(P2 – 2)(P3 – 2)(P4 – 2)(P5 – 2) …… (Pn – 2), the title formula

for Point 4;

4) The ordered set of prims for P!!, or (Pn – 1)!! is (p1, p2, p3, p4, p5, …p, in

numerical order, with p being equal to Pn!! - Pn +1, the (Pn – 1)!!th prim in the set;

5) The “matrix’ for [Pn+1!!] =pp-[Pn+1];

6) The pre-prim set or matrix for [Pn + 1] = pp-[P n + 1]}, and is the table for the set

without C-0.

7) The table for [Pn+1!!] consists of (Pn – 1)!! + 1 columns labeled from left to right

C-0, C-1, C-3, C-4, C-5, ……C-, with each having a value in the uppermost row

of zero, p-1, p-2, p-3, p-4, p-5 ……(Pn!! - 1), respectively, and Pn + 1 rows labeled

from top to bottom R-0, R-1, $-3, R-4 R-5, …… R-Pn +! –1 with each having a

value in the rightmost column of zero, 1(Pn!!), 2(Pn!!), 3(Pn!!), 4(P!!), 5(Pn!!)

……(Pn + 1 - 1 )(Pn!!), respectively;

8) The numerical value of any cell of the matrix is the sum of two numbers, the

number in Row zero of the cell’s row and the number in Column Zero of the

cell’s column, as in an ordinary arithmetical addition table, and as is set out in the

table for (P4 - 1)!! With pp-[P4:] embedded, below;

9) Table for [P4!!] With pp[P4:], the preprim set for 7!! embedded:

0 1 7 11 13 17 19 23 29

30 31 37 41 43 47 49 53 59

60 61 67 71 73 77 79 83 89

90 91 97 101 103 107 109 113 119

120 121 127 131 133 137 139 143 149

150 151 157 161 163 167 169 173 179

180 181 187 191 193 197 199 203 209

10) Each of the 8 strikes in the matrix is underlined and in bold face, and there is one and

only one strike in each column, except for C-0, that is: (7, 49, 77, 91, 119, 133, 161, and

203).

11) Each of the pairs stricken or subtracted from the above preprim set when a strike is

removed is in bold face, and six such pairs are removed: (47, 49), (77, 79), (89, 91),

(119,121), (161, 163), or two for each one of the three twin prime pairs in R-0, which

consists of the ordered set of prims for P3!! including the straddle pair, (29, 31) The first

member of a straddle pair is considered by previous definition to be ‘in” the row of the

first member of the pair,, and the second member of the straddle pair in the first column

of the next row. The last straddle pair in any preprim set is never stricken because (Pn +

1)(Pn!! - 1), the largest strike, equals (Pn + 1)!! - (Pn + 1);

12) Two prim pairs in the above preprim and any larger preprim set are stricken for

each prim pair in R-0, because a pair is stricken when the first of the prims of the pair

is stricken by the strike in one column, and the second prim of the pair is stricken by

the strike in a second non-adjacent column assuming that there is one and only one

strike per column in any preprim set.

13) Since there are 21 preprim pairs, and six preprim pairs are stricken, there are 15

twin prim pairs remaining in (P4 –1)!! as 7 x 3 – 2 x 3 = 5 x 3 = 15, in

accordance with the formula (P4 –2)(P3 – 2)(P2 –2)(P1 – 1) = (7 – 2)(5 – 2)(3 – 2)

(2 – 1) = 5 x 3 x 1 x 1 = 15;

14) There is one and only one strike in each column of a preprim set, pp-[Pn + !]

because:

a. If we let Na and N b be two distinct integers, or zero and an integer from 0

to Pn,

b. and“pc”, and “pd ”, be two distinct prims in R-0 of pp-[Pn + 1]m and “pe”

be a different prim in R-0, and

c. (Pn + 1)(pc) and (Pn + 1)(pd) be two distinct strikes in pp-[Pn + 1], then,

d. If we then let: (Na)(Pn!!) + pe = (Pn +1)(pc), and

(Nb(Pn!!) + pe = (Pn + !)(p d);

e. And then subtract the first above equation from the second above

equation to obtain: (Nb - Na)(Pn!!) = (Pn + 1)(pd - pc), in which the right

hand side of the equation has a prime factor of (Pn + 1) and the left hand

side of the does not, all of its prime factors being less than Pn +. 1;

15) The last equation is contrary to the Fundamental Theorem of Arithmetic, so that

there is one and only one strike in each column of a preprim set.

16) There being one and only one strike in any column of any preprim set, two twin

prim pairs from the previous prim set are stricken for each distinct prim pair in the

previous prim set, as indicated above, and subtracted from the (Pn + 1)([p]n) prim

pairs so that [p]n + ! = (Pn + 1)([p]n - 2([p]n = (Pn + 1 -2)([p]n :

17) Since [p]4 = (P4 - 2)(P3 –2)(P2 - 2)(P1 – 1) = (P4 - 2)!!,

[p] 5 = (P5 – 2)(P4 – 2)(P3 – 2)(P2 – 2)(P1 – 1) = (P5 –2)!!,

[p]6 = (P6 – 2)(P5 – 2)(P4 –2)(P3 – 2)(P2 –2)(P1 –1) = (P6 - 2)!!

[p]n = (Pn – 2)(Pn – 1 –2)(Pn – 3 -2) ……(P3 –2)(P2 – 2)(P1 - 1) = (Pn –2)!!

18) Since “n” may be any integer without upper limit, by the convention of

mathematical induction, where“n” is without upper limit, [p]n = (Pn – 2)!!, the

equation asserted as “POINT 1” is proven for all “n” where “n” is any integer.

POINT 5 EXCEPT FOR “1,”ALL OF THE PRIMS LESS THAN (Pn)2

IN THE PRIM SET FOR [Pn!!] ARE PRIMES.

By definition, none of the prims less than (Pn)2 have factor equal to or less than

than Pn, ergo all of prims less than (Pn)2 in [Pn!!] are primes, except for “1.” (E.g. The set

of prims less than 72 in (P4!! is 1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, each of which

is prime except “1” . All other integers from zero through 49 are nonprims which have a

prime factor of either 2, 3, 5, or 7.

POINT 6 DEFINITIONS AND DISCUSSION OF PRIM AND

NONPRIM RATIOS AND DEVIATIONS

A, Let:

1) x ≡ any integer in [Pn!!];

2) y ≡ a real number from zero through Pn!!;

3) Δ1 ≡a real number form zero to but, not through 1;

4) [p]x ≡ the ordered set of prims from zero through x in Pn!!, or the number of

them;

5) [m]x ≡ the ordered set of nonprims from zero through x in Pn!! or the number of

them;

6) [p]n/Pn!! ≡ the prim ratio for [Pn!!], [p]n being the number of prims from zero

through Pn!!;

7) [m]n/Pn!! ≡ the nonprim ratio for [Pn!!], [m]n being the number of nonprims from

zero through Pn!!;

8) dpx ≡ the “prim deviation” in [Pn!!] at x, which is the absolute number of prims

which which must be added or subtracted from the number of prims from

zero through x to make the resulting prim ratio” at x equal to the prim

ratio for [Pn!!] (A synonym for “deviation” is “amplitude.);

9) If the prim ratio at x is greater than the prim ratio for Pn!!, the prim ratio at x is

said to be positive or positively skewed and │dpx│ must be subtracted

from [p]x so that the resulting prim ratio at [p]x equals the prim ratio for

Pn!!;

10) If the prim ratio at [p]x is less than the prim ratio for Pn!!, the prim ratio is said to

be negative or negatively skewed and │dpx│ must be added to [p]x to

make the resulting prim ratio at x equal to the prim ratio for Pn!!;

11) If the nonprim ratio at x is greater than the nonprim ratio for Pn!!, the nonprim

ratio at x is said to be positive or positively skewed and │dmx│ must be

subtracted from [m] to make the resulting nonprim ratio at x equal to the

nonprim ratio for Pn!!;

12) If the nonprim ratio at x is less than the prim ratio for Pn is less than the nonprim

ratio for Pn!!, the nonprim ratio at x is said to be negative or negatively

skewed and │dmx│ must be added to [m]x to make the resulting nonprim

ratio at x equal to the nonprim ratio for Pn!!;

13) maxposdpx, maxnegdpx, maxposdmx, and maxnegdmx equal the maximum positive prim

deviation at x, the maximum negative prim deviation at x, the maximum positive

nonprim deviation at x, and the maximum, negative nonprim deviation at x respectively,

where x may represent different integers, such as y, z and w, in some of the cases;

B.

1) [p]x + [m]x = x

2) [p]x/x + [m]x/x = 1

3) [p]x/x ± dpx/x = [p]n/Pn!!, the sign being plus if the prim ratio at x is negative, and

minus if it is positive.

4) [m]x/x ± dmx/x = [m]n/Pn, the sign being plus if the nonprim ratio at x is negative , and

minus if it is positive.

5) When the prim ratio at x is positively skewed at x, the nonprim ratio is negatively

skewed at x, when the prim ratio is negatively skewed at x, the nonprim ratio is positively

skewed at x, and the prim and non prim ratios are unskewed at Pn!!.

6) At some integer x in [Pn!!], maxposdpx = maxnegdmx, when maxima are measured at

integers.

7) At some other integer, y, in [Pn!!], maxposdmx = maxnegdpx, when maxima are measured at

integers.

8) The maximum positive dpx is always at a prim and the maximum positive dmx is always

at a nonprim which is the last nonprim in a series of one or more nonprims. (P1!! is a

special case where the maximum positive dmx is zero.)

9) Whereas , for the purposes of this proof the maximum negative and maximum positive

prim and nonprim deviations are always measured at an integer, the deviations

themselves need not be an integer, and the maximum negative prim and nonprim

deviations, if not required to be measured at an integer, are: for the maximum negative

prim deviation, at an infinitesimal before a prim; and for the maximum negative nonprim

deviation at an infinitesimal before the first of as series of one or more nonprims.

10) Although not needed for this proof, it can be shown that: (a) The maximum positive

prim deviation in Pn!! equals the maximum negative prim deviation, at an integer, plus

Δ1, that is maxposdpx = maxnegdpy + Δ1, and (b) The maximum negative nonprim deviation at

some integer in Pn equals the maximum positive integer at some y plus Δ1.

(Except for 2!!, the set of prims in Pn!! is bilaterally symmetrical about Pn!!/2 and

maxposdpx is at some integer x in [Pn!!]. The interval from x plus an infinitesimal to Pn!!,

being bilaterally symmetrical about Pn!!/2 is the same as the prim distribution from zero

through (Pn!!-x), minus an infinitesimal, so that the integer x is not counted twice. This

means that to find the maximum negative prim deviation at an integer, it must be

measured at (Pn!! –x -1 for the denominator and numerator so that the maximum negative

prim deviation is less than the maximum positive deviation by Δ1. Excepting [2!!] the

nonprim set is bilaterally symmetrical about Pn!!, except for the nonprim integer Pn!!,

which has no counterpart at zero, and the maximum positivre nonprim deviation is at a

nonprim x while the maximum nonprim negative deviation is at an infinitesimal less than

the integer (Pn!!-x) to which the integer Pn!!, that must be added, so that maxnegdmy = maxpos

dmx +1 – Δ1. Since Δ1 is some number less than 1, 1 – Δ1 equals some other Δ1 In sum,

when measured at integers, the maximum positive prim deviation equals the maximum

negative prim deviation plus Δ1, and the maximum negative deviation equals the

maximum positive nonprim deviation plus Δ1, when measured at integers.)

POINT 7

THE PRIM AND NONPRIM DEVIATIONS IN [Pn!!] ARE LESS THAN

│.7053│(P

1) The maximum negative prim deviation for [Pn!!] is not more than βn(Pn) + 1,

where βn = 1/P1 + 1/P1xP2 + 1/P1xP2xpP3 + 1/P1xP2xP3xP4 + 1/P1xP2xP3xP4xP5

+…… + 1/P1xP2xP3xP4xP5x ……xPn + ……. When the coefficient of the series

is taken to n terms, it is denominated by “βn.”

2) “βn = 1/P1!! + 1/P2!! + 1/P3!! + 1/P4!! + 1/P5!! + ……1/Pn!! + ……., where “n”

can increase without upper bound. In this formulation, each successive term is

expressed as a reciprocal of the respective ordered primorial numbers from P1!!

onward.

3) Expressed in numbers, βn = ½ + 1/2x3 + 1/2x3x5 + 1/2x3x5x7 + 1/2x3x5x7x11 +

1/2x3x5x7x11x13 + 1/2x3x5x7x11x13x17 + ,,,,,,. + 1/Pn!! +…….

4) For Pn > 7, βn < .7035, which is about .015 less than (e – 2), “e” being 2.

718281824….

5) To demonstrate that the prim deviations for [Pn!!} are not more than 7.035, first

add the integer Pn!! to the positive real number series at zero. (This is consistent

with the numbering system in that each integral multiple of a prime factor in

[Pn!!] is an integral multiple of a corresponding prime factor in the Pn!! placed a

zero. That is, all of the corresponding prime factors in the whole set are congruent

to said prime factor. Also, when Pn!! Is added at zero the nonprims in [Pn!!]

become bilaterally symmetrical about Pn!! with the extra integer included as a

nonprim, and if we consider [Pn!!] to be the cyclic or polar set previously

described as looping back to zero, the use of placing Pn!! at zero is and will be

shown to be consistent with the proof of this point.)

6)

POINT 8 THE TWIN PRIM DEVIATIONS FOR [Pn!!] ARE NOT MORE THAN │Pn│.

At each of the prims in [Pn!!] the prim and nonprim deviations are not more than

│Pn,│ so that the twin prim deviation at each of the twin prim pairs in [Pn!!] is also not

more than │Pn│ twin prim pairs, and further, said twin prim deviations are not more

than J, or than (e-2.) (See Points 6 & 7) Put another way, there can be at most one twin

prim deviation for each prim deviation. (Note that no two twin prim pairs or twin prime

pairs share an integer, except for (3,5), and (5,7).

POINT 9

ф(Pn) IS THE PRIM RATIO FOR [ Pn!!] TIMES (Pn) 2, EXPRESSED IN

MULTIPLES OF Pn AND FOR Pn ≥ 311, ф IS GREATER THAN 30.

1) The prim ratio for [Pn!!] equals:

(1/2)(2/3)(4/5)((6/7)(10/11)(12/13)(16/17)(18/19)(22/23)…..((Pn – 1)/ Pn),

which also can be also be represented by, by transposing each of the

numerators one place to the left as follows:

(2/2)(4/3)(6/5)(10/7)(12/11)(16?13)(18/17)(22/19) …….1/Pn. Since each term

of this expression is greater than 1, except the first, which equals 1, and the

last which is 1/Pn. ф increases both as Pn or (Pn)2 increase .

2) As examples:

a) The prim ratio for [19!!] x 192 is greater than 2.78(19), so that ф for

{19!!} is greater than 2.78.

b) The prim ratio for 311!! Is approximately .096823717, which when

multiplied by (311)2 equals more than (30.11) (311), so that ф for all primes

greater than 311, ф is greater than 30.11. ( P64 = 311))

3) The number of primes from 311 to (311)2, as counted is 9,246, and as

calculated for the average number of prims in the same period, (all of which

are primes, except”1”) the number of primes is 9,365, a difference of

approximately 119. Since the maximum prim deviation is less than

(.7182818210)311)311 or less than 225, the counted number lies within the

number allowed by the prim deviation for the primes subsequent to 311, the

number of primes between them and their squares is at least 29(Pn + x).

4) It can be shown that for each

(Quaere: Does ф increase without upper bound as Pn increases without upper

bound, and, where Pn is sufficiently large, is the prim deviation always positive

thereafter?)

POINT 10

σ(Pn) IS THE TWIN PRIM RATIO FOR [(Pn -2)!!] TIMES

(Pn)2, EXPRESSED IN MULTIPLES OF Pn, AND FOR Pn ≥ 1069,

σ > 8.954.

The twin prim ratio for [(Pn – 2)] is (Pn – 2)!!/Pn, and is:

(1/1)(1/2)(3/5)(5/7)(9/11)(11/13)((15/17)(17/19)(21/23)……….((Pn – 2)) /Pn). By

transposing each of the numerators one position to the left, the twin prim ratio can be

expressed as:

(1/1)(3/2)(5/5)(9/7)(11/11)(15/13)(17/17)(21/19) ………((Pn -2)/Pn-1))(1/Pn. Multiplying

this by Pn gives σ and σ(Pn) is the average number of twin prim pairs in a consecutive

series of (Pn)2 integers in [(Pn – 2)!!].

The calculated twin prim ratio for Pn = 1069 is .008376109…. This twin prim

ratio times (1069)2 equals (8.954…)(1069), so that σ for 1069 is greater than 8.954, that

is σ1069 > 8.954. Since σ increases gradually as Pn increases without limit, σ is always

greater than 8.954 when Pn is greater than 1069, and σ minus the twin prim deviation of

not more than ΕΖ is always greater than 8, when Pn is greater than 1069.(Quaere: Does

sigma increase without upper bound as Pn increases without upper bound and is the twin

prim deviation either always positive or always negative after Pn is sufficiently large.?)

For an example, consider the prime 1007 its square 1,014,049, and the twin prim

ratio for 1007 which is .008576576194…. The latter number times 1,014,049 equals

8,890.87 twin prime pairs between 1007 and 1,014,049 as calculated without

consideration of the deviation. The counted number of twin primes between 1007 and

1,014,049 is 9,508. The difference between the counted number of pairs and the

calculated number of pairs is (9508 – 8,890) or 618, the counted number of pair lies

within the allowable deviation of 710 twin prim pairs, which is €Ζ(1007) or

(,70517943…)(1007), (Note that the twin prime deviation is positive.)

POINT 11

THERE IS AN INFINITE NUMBER OF TWIN PRIMES.

Since, beyond 1069, as Pn increases without limit, there are always at least 8Pn

twin primes between Pn and its square, by the convention of mathematical induction,

there is an infinite number of twin primes!!

ADDENDUM 1

A FORMULA FOR THE NUMBER OF PRIMES FROM ZERO THROUGH

(Pn)2 is: [P]p^2 = {[(Pn -1)!!] ÷ Pn!!} x (Pn)2) + (n-1) ± Pn(β), THE TERM ‘(n-1)’

BEING THE NUMBER OR PRIMES FROM ZERO THROUGH Pn, MINUS “1”,

WHICH IS A PRIM BUT NOT A PRIME, AND “Δ’ BEING LESS THAN │1│.

The above formula follows from the foregoing proof. The formula for the number

of primes from zero through (Pn)2 given by the Prime Number Theorem is:

[P]P^2 ≈ (Pn)2/ln(Pn)

2. The non-analytical prim generated formula of “Addendum 1” may

be a useful adjunct to the Prime Number Theorem.

Since the terms “(n-1)” and “Pn(β) become vanishingly small as (Pn)2 increases without

limit, the formula: {[(Pn – 1)!!] ÷ Pn!!} x ((Pn)2)) ≈ [P]Pn^2/ ln(Pn)2, as Pn increases

without upper bound. These formulae can be used for approximations of ln(Pn)2.

ADDENDUM 2

A FORMULA FOR THE NUMBER OF TWIN PRIMES FROM ZERO

THROUGH (Pn)2, AS Pn INCREASES WITHOUT UPPER BOUND, IS:

[tp]p^2 ≈ {[(P – 2)!!] ÷ Pn!!} X (Pn)2.

The above follows from the foregoing proof and Addendum 1.

ADDENDUM 3

A FORMULA FOR THE FINE STRUCTURE CONSTANT

Let:

1) P = any prime;

2) n = the ordinal number of a prime;

3) Pn = the nth prime;

4) Pn!! = the product of the first n primes or primorial Pn;

5) Let (Pn – 1)!! = the product of the first n primes with each prime reduced by ,

or: (2 – 1)(3 – 1)( 5 – 1)(7 - 1)(11 – 1))13 – 1)(17 – 1)(19 – 1)(23 – 1)(29 –

1)….(Pn – 1). This is the expression for the number of integers from zero

through Pn!!, called prims, which do not have any of the prime factors of Pn as

a factor. (Note that P10 = 29.);

6) α-1 = 137.0359990710(96), the reciprocal of the fine structure constant;

7) Bn ≡ [(P3!!2

- Pn!!)(P10 – 1)!!] ÷ P10!! = [(2.3.5)2 – 2.3.5] x (2 -1)(3-1)(5 – 1)(7

-1)(11-1)(13 – 1)(17 -1)(19 -1)(23 – 1)(29 – 1) ÷

2x3x5x7x11x13x17x19x23x29;

8) Bn = 870 x [(P10 – 1)!! ÷ P10!!] = 870 x .1594722310194…;

9) Bn =137.41490343241359…, rational number;

10) mp =1836.152627161…, the rest mass of the proton, as measured. with a

statistical uncertainty of .0000085;

11) mn = 1838.68366598 the rest mass of the neutron, as measured, with a

statistical uncertainty of .000013

12) (mp / mn)2 = .99772488033641.,

13) Bn(mp / mn)2 = 137.036145129…;

14) Bn(mp / mn)2minus α-1 is (137.036145129 – 137..035999070(96) =

.0000261…. or approximately 1.8 parts in 10 million, whereas the measured

value is for α-1 is said to be accurate within .7 parts per billion provided by

the formulaic value;

15) Bn(mp /mn )2 ≈ α-1.

16) The statistical uncertainties for the measured values of mn and mp are

.0000085 and .00085 respectively, so that the formula “Bn(mp / mn)2” is an

equality through five significant figures and a possible equality thereafter.

The plausibility of the above formula is suggested by the following:

1) It seems to spring naturally from an attempt to prove the Twin Prime Theorem

using the prim system of numbers inspired by the use of Euler’s Totient;

2) At least two number theorists have found the prime”29” of use in constructing

possible numerical representations of the fine structure constant;

3) A recent measurement of the Fine Structure Constant involving 891 four loop

Feynman Diagrams suggests a slight association with the number “870” in the

formula;

4) “30” is an important number as it is the largest primorial number for which all

of the prims, except “1, are prime”;

5) The prim and nonprim sets for a primorial numbers are roughly analogous to

the bandings of various spectra of light;

6) The prominence of the number 2(2.3.5)2 suggests: mp ≈ 2(2.3.5)2 + (2.3)2 =

1836.

7) The fact that the formulaic value for α-1, differs from the measured value by

only about 1,8 parts in ten million, which is within the allowable statistical

uncertainty for mp, suggests that more may be at work in this matter than mere

coincidence.

ADDENDUM 4

mp/mn ≈ (α-1/Bn)1/2, AND mp ≈ (1/αBn)1/2mn.

The above formulae are derived by rearranging formulae in Addendum 3, and indicate

the relationships among the rest masses of the proton, the neutron, the electron, given as

“1”, the Fine Structure Constant, and Bn.

By inserting the numbers for some of the respective letters in the above formula

for mp, we obtain: mp ≈ [((α-1)(29!!)) / (29 + 1)(29)(29-1)!!]1/2 x mn, a remarkable result.